L(s) = 1 | + 2·16-s − 8·31-s − 4·41-s + 8·61-s − 16·71-s + 81-s − 8·101-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | + 2·16-s − 8·31-s − 4·41-s + 8·61-s − 16·71-s + 81-s − 8·101-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07132271713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07132271713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T^{4} + T^{8} \) |
good | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 13 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 17 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | \( ( 1 + T + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 43 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 53 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 59 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 61 | \( ( 1 - T + T^{2} )^{8} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 + T )^{16} \) |
| 73 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 89 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.05167961364493317466960213993, −4.03797993435311022904855569012, −3.92881235075408855005909353873, −3.91792195142930063995557640940, −3.91134506199235148467388215251, −3.70719740993966932509469020224, −3.38515846241032667959857457831, −3.16003554654582166684090138636, −3.15398381557730017306994834060, −3.09969212426919849260878773271, −2.99856500217957436810707241741, −2.93760197554213190169260213367, −2.93678566040578532679537013896, −2.64313276421559857091952309918, −2.27961541601106450202869188209, −1.98491876706737872351358226189, −1.96924782360785571842396142702, −1.90488381638609017449048181607, −1.80503919206670076590381488672, −1.62785111165918555781214206912, −1.57445781972414155785619589187, −1.25884491901131651590652254621, −1.23447959103052203842511443275, −0.855545169668131933687633899687, −0.11204262707789757538186926695,
0.11204262707789757538186926695, 0.855545169668131933687633899687, 1.23447959103052203842511443275, 1.25884491901131651590652254621, 1.57445781972414155785619589187, 1.62785111165918555781214206912, 1.80503919206670076590381488672, 1.90488381638609017449048181607, 1.96924782360785571842396142702, 1.98491876706737872351358226189, 2.27961541601106450202869188209, 2.64313276421559857091952309918, 2.93678566040578532679537013896, 2.93760197554213190169260213367, 2.99856500217957436810707241741, 3.09969212426919849260878773271, 3.15398381557730017306994834060, 3.16003554654582166684090138636, 3.38515846241032667959857457831, 3.70719740993966932509469020224, 3.91134506199235148467388215251, 3.91792195142930063995557640940, 3.92881235075408855005909353873, 4.03797993435311022904855569012, 4.05167961364493317466960213993
Plot not available for L-functions of degree greater than 10.